2D Materials 2D Materials Polaritonics Polaritonics 2D ...
Transcript of 2D Materials 2D Materials Polaritonics Polaritonics 2D ...
2D Materials 2D Materials PolaritonicsPolaritonics-- QuickQuick tutorial tutorial --
2D Materials 2D Materials PolaritonicsPolaritonics-- QuickQuick tutorial tutorial --
Tony LowUniversity of Minnesota, Minneapolis, USA
Email: [email protected]: http://people.ece.umn.edu/groups/tlow/
IMA UMN, 6-10th Feb 2017
About us
Nanoelectronics Nanophotonics
Multiphysics and multiscale modeling of 2D materials electronics and photonics for computing and communication devices.
Mission:
• 2D materials polaritonics• Photodetectors• Reflectarray• Modulators• Sensors
• 2D materials and transport physics• Tunneling devices• Spintronics• Valleytronics• Strain and piezoelectronics
2
- ++
+ +
---
- ---
Graphene
plasmon
Polaritons – marrying the best of both worlds
-+ -
+ -+
Transition metal
dichalcogenides
exciton
3
Quick overview
Basics on graphene plasmonsA pedagogical tutorial on graphene plasmonics starting from Maxwell eq.
Graphene plasmonics Graphene plasmonics A review on graphene plasmonics experiments and its applications
Beyond graphene plasmonicsA forward looking perspective on what’s new with other 2D materials
4
Polaritons in 2D materials
Graphene Boron nitride Transition metal dichalcogenides
T.Low, J.Caldwell, F. Koppens, L.M.Moreno, P. Avouris, T. Heinz et al, Nature Materials (2016)
5
Plasmon as collective electronic excitations
External perturbation screened within a Thomas Fermi length
0
( )( ) 1 ( ) 1
i
σ ωε ω χ ωωε
= + = + ( ) 0plε ω ω= =
Collective electronic oscillation i.e. plasmons
2D materials carrier concentration tunable up to 0.01 electrons per atom THz and mid-IR plasmon 6
Technologies across electromagnetic spectrum
Terahertz to Mid-infrared Contains atmospheric transmission window Super high-speed wireless communication Imaging for military, security & medical Detections of molecules for bio. and chem.
7
Possible applications for graphene plasmonics
IBM, Nature Nano (2012)
EPFL, Science (2015)
IBM, Nature Com (2013)
IBM, Nature Nano (2012)
U Penn, Science (2012)
Far field communications, e.g. modulator, reflectarray for far-field MIR
8
Applications of polaritons in 2D materials
T.Low, J.Caldwell, F. Koppens, L.M.Moreno, P. Avouris, T. Heinz et al, Nature Materials (2016) 9
Quick overview
Basics on graphene plasmonsA pedagogical tutorial on graphene plasmonics starting from Maxwell eq.
Graphene plasmonics Graphene plasmonics A review on graphene plasmonics experiments and its applications
Beyond graphene plasmonicsA forward looking perspective on what’s new with other 2D materials
10
( , )( , )
B r tE r t
t
∂∇× = −∂
Faraday’s Law
( , )( , ) ( , )
D r tH r t J r t
t
∂∇× = +∂
Ampere’s Law
( , ) 0, ( , ) ( , )B r t D r t r tρ∇ ⋅ = ∇ ⋅ = Gauss’s Law
Maxwell equations in SI units
Constitutive relations
Maxwell equations in a nutshell
Constitutive relations
( , )
( , ) 0r t
J r tt
ρ∂ + ∇ ⋅ =∂
( , ) ' ( ', ') ( ', ')
( , ) ' ( ', ') ( ', ')
( , ) ' ( ', ') ( ', ')
t
t
t
D r t dt dr r r t t E r t
B r t dt dr r r t t H r t
J r t dt dr r r t t E r t
ε
µ
σ
−∞
−∞
−∞
= − − ⋅
= − − ⋅
= − − ⋅
∫ ∫
∫ ∫
∫ ∫
Fields are related by permittivity,
permeability, conductivity tensors
Continuity equation
Ohm’s law11
Assumptions
• Mediums are not spatially dispersive, i.e. local response
• Linear response, monochromatic waves
• No free charges or currents
( , ) ( ) ( , )E r i H rω ωµ ω ω∇× = ⋅
( , ) ( , )exp( )E r t E r i tω ω→ −
( , ) ( ) ( , )H r i E rω ωε ω ω∇× = − ⋅
( ) ( , ) 0E rε ω ω∇ ⋅ ⋅ =
Maxwell equations Boundary conditions
( ) 0n j i
e E E× − =
( ) ( , )n j i
e H H J r ω× − =
Current at interface to be included i.e.
where 2D materials is also described!
Maxwell equations in a nutshell
( ) ( , ) 0E rε ω ω∇ ⋅ ⋅ =
( ) ( , ) 0H rµ ω ω∇ ⋅ ⋅ =
( , ) ( ) ( , )D r E rω ε ω ω= ⋅ ( , ) ( ) ( , )B r H rω µ ω ω= ⋅
Constitutive
where 2D materials is also described!
12
( , ) ( ) ( , )E r i H rω ωµ ω ω∇× = ( , ) ( ) ( , )H r i E rω ωε ω ω∇× = − ( , ) 0E r ω∇ ⋅ =
Maxwell equations
In the isotropic case,
2
2
( , ) ( ) ( , ) ( ) ( ) ( , )
( ( , )) ( , ) ( ) ( ) ( , )
E r i H r E r
E r E r E r
ω ωµ ω ω ω µ ω ε ω ωω ω ω µ ω ε ω ω
∇×∇× = ∇× =∇ ∇ ⋅ − ∆ =
Maxwell equations in a nutshell
( , ) 0E r ω∇ ⋅ = ( , ) 0H r ω∇ ⋅ =
( , ) ( ) ( , )D r E rω ε ω ω= ( , ) ( ) ( , )B r H rω µ ω ω=
Constitutive
2
( ( , )) ( , ) ( ) ( ) ( , )
( , ) ( ) ( ) ( , )
E r E r E r
E r E rω ω µ ω ε ω ω∆ = −
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Solution
0( , ) ( , )exp( )E r E k ik rω ω= ⋅
2 2
22 2 2
2
( , ) ( ) ( ) ( , )
( ) ( )
k E r E r
k kv
ω ω µ ω ε ω ωωω µ ω ε ω
− = −
− = − ⇒ =
2exp( ) exp( )ik r k ik r∆ ⋅ = − ⋅Using
x
z
1ε
2ε
We are interested in finding EM modes localized at the interface, 2D material
| |E
z
x zk e ieβ γ= ±
This localized EM mode is reflected in the following ansatz
0 exp( )exp( ) for 0( , )
exp( )exp( ) for 0
A i x z zA r t
A i x z z
β γβ γ
− >= <
Confined EM modes, TE plasmons
0 exp( )exp( ) for 0A i x z zβ γ <
We start with the electric field for the TE plasmons
1 1 1 1( ) ( , ) exp( )exp( ), 0y y
E r e E x z e E z i x zγ β= = <
2 2 2 1( ) ( , ) exp( )exp( ), 0y y
E r e E x z e E z i x zγ β= = − >
2 2
0where j j
γ β ω µ ε= −
The magnetic field takes the form,
( )1 11 1 1
1 1
( , ) ( , )1 1( , ) ( , )
z x z x
E x z E x zH x z e e e i e E x z
i x z iβ γ
ωµ ωµ∂ ∂ = − = − ∂ ∂
( )2 2 2
2
1( , ) ( , )
z xH x z e i e E x z
iβ γ
ωµ= +
14
From boundary conditions
1 2E E=
2 12 1 1
0 0
E E i Eγ γ σωµ µ
+ =
We obtain the solution for electric field
11 2 0
0
( ) 0E
iγ γ σωµµ
+ − =
Which has non zero solution if,
Confined EM modes, TE plasmons
1 2 0 0iγ γ σωµ+ − =This is also the pole of the Fresnel coefficients for TE waves!
We can obtain plasmon dispersion in free-standing case,
2 2 2 2 22 2 2 20 0
0 0 0 0
0 0 0 0 0 0
2 2
00
2 14 4
where , . Thus the TE plasmon is
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i k k
k
k
σ ω µ σ ηγ σωµ β β
ω ε µ η µ ε
σ ηβ
= → − = − → = −
= =
= −15
x zk e ieβ γ= ±
This localized EM mode is reflected in the following ansatz
0
0
exp( )exp( ) for 0( , )
exp( )exp( ) for 0
A i x z zA r t
A i x z z
β γβ γ
− >= <
We start with the magnetic field for the TM plasmons
1 1 1 1
2 2 2 2
( ) ( , ) exp( )exp( ), 0
( ) ( , ) exp( )exp( ), 0
y y
y y
H r e H x z e H z i x z
H r e H x z e H z i x z
γ βγ β
= = <
= = − >
2 2
0where j j
γ β ω µ ε= −
The electric field takes the form,
Confined EM modes, TM plasmons
( )
( )
1 1 1
1
2 2 2
2
1( , ) ( , )
1( , ) ( , )
z x
z x
E x z e i e H x zi
E x z e i e H x zi
β γωε
β γωε
= − −
= − +
From boundary conditions
1 2 11 2 2 1 1
1 2 1
2 1 1 1 21 1 2 2 1
1 2 1
0,
we obtain 1 0 0
H H H H Hi
Hi i
γ γ γσε ε ωε
ε γ γ γ γσ ε γ ε γ σε γ ωε ω
+ = − = −
+ − = → + − =
16
We can obtain plasmon dispersion in free-standing case,2 2 2
2 20 00 0 0 0 0
0 0 0 0 0 0
0 2 2
0
42 0 2
where , . Thus the TM plasmon is
4 1
i ki
k
k
σγ ε ωε γ γ ε ω σ βω σ
ω ε µ η µ ε
βσ η
− = → = → − = −
= =
= −
This is also the pole of the Fresnel coefficients for TM waves!
Confined EM modes, TM plasmons
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The equation of motion of free electrons in metal
electron
momentum
relaxationtime
2
, we also have
Hence,
p mvp dpeE
J envdt
dJ J neE
dt m
τ
τ
=− − =
= −
+ =
Assuming time dependence exp( ) , exp( ), we obtain, E E i t J J i tω ω= − = −
Drude conductivity
0 0
2 2
0 0
2
2
Assuming time dependence exp( ) , exp( ), we obtain,
( 1 ) ( )
To map the relation to graphene, we use the relation,
and
then
F F
F
E E i t J J i t
ne ne ii J E
m m i
E km n
v
ω ω
ω τ σω τ
π
= − = −
− + = → =+
= =
2
2
,
where (also known as Drude weight)( )
Fe EiD
Di
σω τ π
= ≡+ ℏ
18
2 2
00
0 2 2
0
TE plasmons, 14
4TM plasmons, 1
k
k
σ ηβ
βσ η
= −
= −
2
2 where
( )
Fe EiD
Di
σω τ π
= ≡+ ℏ
η µ ε σ
η σ
−= = Ω = ×∼ℏ
∼ ≪
25
0 0 0
Lets consider some typical numbers,
376.6 and | | 6.1 104
Hence, | | 0.02 1. This implies that,
eS
Graphene plasmons
η σβ
ω ωπ π ε ωβση η η
=
=
→
∼ ≪
ℏ ℏ∼ ∼ ∼
0
0
2 2 2
0 0 00 2 2
0 0 0
Hence, | | 0.02 1. This implies that,
TE plasmons,
2 2 22TM plasmons,
F F
k
k kik
D e E e E
βω βπ ε ε
= =ℏ
2
2
0 0
2 2
F
pl
e E D
19
Quick overview
Basics on graphene plasmonsA pedagogical tutorial on graphene plasmonics starting from Maxwell eq.
Graphene plasmonics Graphene plasmonics A review on graphene plasmonics experiments and its applications
Beyond graphene plasmonicsA forward looking perspective on what’s new with other 2D materials
20
E Light
Tera
hert
z to
Mid
-IR
Maxwell
Exciting plasmons in graphene
0.2eV+ -+
+
--
E
2
202pl
r
e qωπ
µε ε
=ℏ
q
Tera
hert
z to
Mid
M.Jablan et al, Phys. Rev. B (2009)F. Koppens et al, Nano Lett. (2011)
L.Ju et al, Nature Nano (2011)H.Yan et al, Nature Nano (2012)
J.Chen et al, Nature (2012)Z.Fei et al, Nature (2012)
T.Low and P.Avouris, ACS Nano (2014)H.Yan, T.Low et al, Nature Phot.(2013)
+
+
---
Understand Graphene plasmonic resonator and what we can do with it
21
ener
gy
RPA Loss function1( , ) Im RPAL q ω ε − =
2
202pl
r
e qµωπ ε ε
=ℏ
Mid-infrared plasmons with graphene nanostructures
1 2~ , ( tan [ 4 / (4 )])RR
nq
W
π π π−− Φ Φ = − +
momentum
A.Y.Nikitin, T.Low, L.M.Moreno, PRB Rapid (2014)
H.Yan, T.Low et al, Nature Photonics (2013)T.Low and P.Avouris, ACS Nano (2014)
( , ) 1 per
Par
TZ W
Tω = −
Measuring extinction:
22
8
12
16
20Optical Phonon
Peak 3Peak 2
Width (nm) 60 70 85 95 100 115 125 140 150 170 190 240
1-T
per/T
// (%
)
2nd order
Peak 1
1000
1500
2000
2500
ωsp2
Wav
e nu
mbe
r (c
m-1)
ωop
Graphene on SiO2
First peak Second peak Third peak 2nd order mode ω ~ q1/2
Mid-infrared plasmons with graphene nanostructures
1000cm-1 ~ 10um ~ 30THz
1000 1500 2000 2500 3000
0
4
Wave number (cm-1)0 2 4 6 8
0
500
1000ω
sp1
Wav
e nu
mbe
r (c
m
Wave vector q (x105 cm-1)
H.Yan, T.Low, F.Guinea et al, Nature Phot. (2013)T.Low and P.Avouris, ACS Nano (2014)
Plasmon dispersion can be engineered with substrates
23
( ) ( ) ( )
( )
βε β ω β ω β ωβε π ω τ
ε β ω
= − ∏ = ∏ =+
=
ℏ
22
2 2
0Coulomb Polarizabilitypotential
Dielectric function of graphene (free electron contributions only)
, 1 , where and ,2 ( )
Plasmon occurs when , 0, ignori
F
c c
Eev v
i
ββε π ω
ββε ω
→ = −
→ = −
ℏ
22
2 2
0
2 2
2 2
ng damping, we have,
0 12
0 1
FEe
e D
Graphene plasmons
( )
βε ωβ βω ωε ε
ωε β ω
ω τ
→ = −
→ = → =
= −+
2 2
0
2
0 0
2
2
0 12
2 2
Hence, we can also express dielectric function as,
, 1( )
pl pl
pl
e
D D
i
24
( )
( )
( )
αε ω ωω τ ω
ω αε β ωω τ ω τ ω
ε β ω
=+ −
= − −+ + −
=
02 2
0
2
2 2 2
0
Dielectric function of polar phonons can be described by
where is phonon frequency( )
Total dielectric function becomes
, 1( ) ( )
Hybrid modes when , 0, assu
pl
i
i i
ω α− − =2
ming again zero damping,
1 0pl
Plasmons and phonons hybridization
ω αω ω ω
ω ω ω ω ω ω αω
ω ω ω ω ω ω ω αω
ω ω ω ω α ω ω
ω ω α ω ωω ω αω
− − =−
− − − − =
− − + − =
− + − + =
+ − −+ −= ±
2 2 2
0
2 2 2 2 2 2 2
0 0
4 2 2 2 2 2 2 2
0 0
4 2 2 2 2 2
0 0
2 2 2 2 22 20 002
1 0
( ) ( ) 0
0
( ) 0
( ) 4
2 2
pl
pl
pl pl
pl pl
pl plpl
25
y
x
kx
ky
K
K’
e
ky
E3x106 ms-
13x106 ms-1
LightDirac e100 x
100 xSound Vel.
op BE k T≫
Graphene, Dirac electrons
kx
Highest mobility µ=1x106 cm2Vs-1
D.C.Elias et al, Nature Physics (2011)
Quantum Hall effect at Room Temp.
K.S.Novoselov et al, Science (2007)
2
4
e
ℏ
( )Re σ ω
ωFrequency
2µ
VisibleNear-IRMid-IR
1 2 3
Terahertz
απ=2% light absorption
Graphene absorption spectrum
µ
Intraband
Z.Q.Li et al, Nature Physics (2008)
R.R.Nair et al, Science (2008)
Disorder-
mediated
Interband
1 2 3
1eV ~ 8000cm-1 ~ 1.25um ~ 240THz 27
x
z
i
E
rE
t
E
1ε
2ε
We seek the scattering coefficients (transmission, reflection), due to a 2D material at z=0
We seek the scattering coefficients (transmission, reflection), due to a 2D material at z=0
for plane monochromatic waves i.e.
Fresnel coefficients
28
0( , ) exp( )A r A ik r i tω ω= ⋅ −
for plane monochromatic waves i.e.
Without loss of generality, assume xz being plane of incidence2 2 2 where | | and
x zk e e k k kβ γ ω εµ β γ= + = = + =
1tan ( )
βθ γ−=
Angle of incidence
Any EM waves can be expressed as linear superposition of TE and TM waves.
TE:
TM:
y
y
E e E
H e H
=
=
Incident EM wave gives rise to reflected wave in medium 1 and transmitted wave in
medium 2
1 1 1 1( ) ( , ) [ exp( ) exp( )]exp( )y y i r
E r e E x z e E i z E i z i zγ γ β= = + −
2 2 2( ) ( , ) exp( )exp( )y y t
E r e E x z e E i z i zγ β= =
The magnetic fields can be obtained from Maxwell equation i.e. Faraday’s law
1 1 1 1 1 1
0
1( ) [ ( ) ( exp( ) exp( ))exp( )]
z x i rH r e E r e E i z E i z i xβ γ γ γ γ β
ωµ= − − −
1
Fresnel coefficients, TE waves
2 2 2 2
0
1( ) [ ( ) exp( )exp( )]
z x tH r e E r e E i z i xβ γ γ β
ωµ= −
Using boundary conditions
2 1 1 2( ) 0 ( ,0) ( ,0)z
e E E E x E x× − = → =
2 1 1 2 1 2( ) ( ,0) ( ,0)z y x x
e H H e E x H H E xσ σ× − = → − =
Then,
i r tE E E+ =
2 1
0 0
( )t i r t
E E E Eγ γ σωµ µ
− + − =29
Then transmission and reflection coefficients are
1
1 2 0
2t
i
Et
E
γγ γ σωµ
= =+ +
1 2 0
1 2 0
r
i
Er
E
γ γ σωµγ γ σωµ
− −= =+ +
We measure reflection and transmission probably with respect to energy carried by the
EM wave, using the Poynting vector,
2 21 11 1 1
0 0
1 1( 0) Re[ ( ( ) ( ))] Re[( ) ( ) ] | | | |
2 2 2z i r i r i r i r
S z e E r H r E E E E E E S Sγ γ
ω µ ωµ
∗∗ ∗ ∗
= = ⋅ × = − + − − = − = −
Re[ ]1 1 γ γ∗∗ ∗
= = ⋅ × = − = =
Fresnel coefficients, TE waves
22 22 2 2
0 0
Re[ ]1 1( 0) Re[ ( ( ) ( ))] Re[ ] | |
2 2 2z t t t t
S z e E r H r E E E Sγ γ
ω µ ωµ
∗∗ ∗
= = ⋅ × = − = =
Then, the probabilities are
22
2
| || |
| |
r r
i i
S ER r
S E= = = 22
1
Re[ ]| |t
i
ST t
S
γγ
= =
Total internal reflection,
2 2
1 2 2 2 2 and is pure imaginary, 0k k Tε ε β γ β> > → = − =
30
α βσ ωω τ
−= −
− + + −
= +
∑
ℏℏ
'2
, ', , ' ' '
Optical conductity can be computed from the Kubo formula,
', ' , , ', '( ( ')) ( ( ))( , )
( ') ( ) ( ) ( ') ( )
where ' , as is due to ph
j j
k k j j j j j j
k j v k j k j v k jf E k f E kq ie
E k E k i E k E k
k k q q∼
oton momentum.
In this work, we are interested in the local limit, i.e. 0q
−
Graphene low energy Hamiltonian at point,
0
K
q iq
Optical conductivity of graphene
σ−
= ⋅ = +
= ± Ψ = ± +
ℏ ℏ
ℏ, ,
0q
0
with the following eigenvalues and eigenfunctions,
11 and
( )2
we also have the veloci
x y
F F
x y
c v F c v
x y
q iqH v v
q iq
E v qq iq q
σ σ= =
= = − = −, , ,
ty operators
and
and and
x F x y F y
yx x
x cc F x vv F x cv F
v v v v
ikk kv v v v v v
k k k31
α βσ ωω τ
π ω τ
−= −
− + + −
+ −= −+
∑
ℏℏ
'2
, ', , ' ' '
22
2
Consider intraband contributions in conduction band
', ' , , ', '( ( ')) ( ( ))(0, )
( ') ( ) ( ) ( ') ( )
( ( 0)) ( ( ))4
4 ( )
j j
k k j j j j j j
c c
c
k j v k j k j v k jf E k f E kie
E k E k i E k E k
f E k f E kied k
i E
π
π ω τ
θ θπ ω τ π ω τ
∞ ∞
∂= − + ∂+ −
∂ ∂ = − = − + ∂ + ∂
∫ ∫
∫ ∫ ∫
ℏ
2 2 22 2
, 2 2
2 2 22
2
2 20 0 0
( ( ))( )
( )( 0) ( )
( ) ( )cos
( ) ( )
cF x
x cc
cc
cF
c
f E kie v kv k d k
i E kk E k
f Eie v ie f Ed dkk dEE
i E i E
Optical conductivity of graphene
σ ωπ ω τ
−∞ ∂ = − + ∂ ∫ℏ
2
2 0
Straightforward to show that contributions from valence band
( )(0, )
( )
ie f EdEE
i E
σ ωπ ω τ π ω τ
π ω τ
∞ −∞ ∞
−∞
∂ ∂ ∂ = − + − = − + ∂ ∂ + ∂
= +
∫ ∫ ∫ℏ ℏ
ℏ
2 2
2 20 0
2
2
Thus the total intraband conductivity is
( ) ( ) ( )( )
( ) ( )
2ln 2cosh
( ) 2
B F
B
ie f E f E ie f EdEE dEE dE E
i E E i E
ik Te E
i k T
32
σ ωπ ω τ
= +
≈ →
ℏ
≫
2
2
Thus the total intraband conductivity is
2( ) ln 2cosh
( ) 2
For low temperture, , we can simplify further. In this case,
1cosh exp ln 2cosh
2 2 2
B F
B
F B
F F
B B
ik Te E
i k T
E k T
E E
k T k T
σ ωπ ω τ
≈
=+ℏ
2
2
2 2
Hence, we obtain, ( )( )
F F
B B
F
E E
k T k T
ie E
i
Optical conductivity of graphene
π ω τ+ℏ ( )i
ω τσ ωπ ω τ π ω τ
σ ω
∞ + − −= + + + −
≈
∫ℏ ℏ
≫
22
2 2 2 20
2
Including both intraband and interband conductivity, we get,
( )2 ( ) ( )( ) ln 2cosh
( ) 2 ( ) 4
For low temperture, , we can simplify to,
( )
B F
B
F B
ie iik Te E f E f EdE
i k T i E
E k T
ie ω τπ ω τ π ω τ
− ++ + + +
ℏ
ℏ ℏ ℏ
2
2
2 ( )ln
( ) 4 2 ( )
FF
F
E iE ie
i E i
33
Universal optical conductivity
[ ]
2 2
0 0
0 0 0 0
0 0
2
0 0
The absorption is defined as
1
where
2and
2 2
and we can show that
4( ) Re
2
A T R
T R
A
γ σωµγ σωµ γ σωµ
γ ωµω σγ σωµ
= − −
−= =+ +
=+
Nair et al, Science (2008)
34
ℏ
ℏ ℏ
ℏ
ℏ ℏ ℏ
2
2 2
Note for high frequency,
2 ( )( ) ln
4 2 ( )
2 ( )Re[ ( )] Im ln
4 2 ( ) 4
F
F
F
F
E iie
E i
E iie e
E i
ω τσ ωπ ω τ
ω τσ ωπ ω τ
− +→ ∞ ≈ + +
− +→ ∞ ≈ = + +
[ ][ ] ℏ
0 0
0
2
0
2
For normal incidence and high frequencies,
Re 0.022
where 377 and Re4
A
e
γ σωµ
η σ
η σ
+
= ≈
≈ =
Photocurrent mechanisms
Bipolar junction
PVI en µξ∗≈
( )I S S Tσ δ∗≈ −
Visible light
Unipolar junction
PVI en µξ∗≈
1 2( )PTEI S S Tσ δ∗≈ −
1 2( )PTEI S S Tσ δ∗≈ −
N.Gabor et al, Science (2012)35
Photoconductivity experiment
M.Freitag, T.Low et al, Nature Phot.(2013) 36
Bolometric vs photovoltaic
0GV ≈PCI DCI
Photovoltaic
0GV ≫
PCI DCI
Bolometric
M.Freitag, T.Low et al, Nature Phot.(2013) 37
Light
Non-radiative decay < ps
Mid infrared plasmons Thermal photo-response Room temperature operation
Mid-infrared photodetector
Key Ingredients
M.Freitag, T.Low et al, Nature Comm. (2013) M.Freitag, T.Low et al, Nature Photonic (2013)
S-pol
P-pol
Drives a bolometric current
38
-40 -20 0 20 400.0
0.5
1.0
1.5
2.0
W=140nm
Loss
Fun
ctio
n (a
.u.)
Gate Voltage VG (V)
1000
1500
2000
100
160
140
120
ωsp1
ωsp2
W
ave
num
ber
(cm
-1)
ωop
ωexp
W =
200
nm
CO2
Mid-infrared photodetector
G
0 2 4 60
500
ωsp1
Wav
e nu
mbe
r (c
m
Wave vector q (x105 cm-1)
Intraband Landau Damping
M.Freitag, T.Low et al, Nature Comm. (2013)
Gate tunability, thanks to hybrid plasmon-phonon polariton
E
E
x15
39
Driven mechanical oscillator Credit: MIT TechTV
Resonator can acquire a phase from its driving force which is determined by its detuning from resonance 40
θ(V)
Light
Vg=1V Vg=3V Vg=4V
Electrically controlled terahertz and mid-infrared beam reflectors
θi
Mid-infrared light bending
V
W1 WN
θ(V)
Graphene
SiO2
High-κ dielectric
Metal Reflector
C.Eduardo, T.Low, et al, Nanotechnology (2015)
A. Nemilentsau, T.Low, arXiv:1610.05236 (2016)
0
1
sin( ) sin( )2
r i
d
dx
λ φθ θπ ε
− =
Generalized Snell’s law
21 2
2
3tan
8
e
W
ω µφ ωτ ε
− ≈ −
ℏ
41
Graphene plasmonics for THZ and MIR applications
T.Low and P.Avouris, ACS Nano (2014) 42
Quick overview
Basics on graphene plasmonsA pedagogical tutorial on graphene plasmonics starting from Maxwell eq.
Graphene plasmonics Graphene plasmonics A review on graphene plasmonics experiments and its applications
Beyond graphene plasmonicsA forward looking perspective on what’s new with other 2D materials
43
A new class of 2D crystals
Eg=0 eV Eg=6.0 eV
TMOs (Transition Metal Oxides): MoO3, LiCoO2
Graphene Boron Nitride
TMDs (Transition Metal Dichalcogenides): MoS2,WS2,NbSe2 (MX2)
III-VI/V-VI Compounds (Ga,In)2Se3, Bi2(Se,Te)3
Strong in-plane bonds Weak van der Waals interlayer coupling Surfaces ideal self-passivation, intrinsically
good electrical properties Pathway for large scale growth Full range of material properties Black Phosphorus 44
Polaritons in 2D materials
Graphene Boron nitride Transition metal dichalcogenides
T.Low, J.Caldwell, F. Koppens, L.M.Moreno, P. Avouris, T. Heinz et al, Nature Materials (2016)
45
Mid-infrared plasmons with near field optical micro scopy
J.Chen et al, Nature (2012)Z.Fei et al, Nature (2012)
Complex wavevector q of plasmon can be measured as function of frequency ω
Re[ ]qπ
exp( Im[ ] )q x−
Spacing between fringes
Exponential decay of fringes
Figure of merits
A.Woessner et al, Nature Mat (2015)
Figure of merits
Im[ ] Re[ ]q qγ =1 2γ π− Number of cycles plasmon propagates
before amplitudes decay by 1/e
0Re[ ]q kβ =
Damping
Confinement Light confinement by the polariton mode
Current state-of-the-art, γ-1 >25 and β~150
46
Mid-infrared plasmons with near field optical micro scopy
bilayer
monolayer
Gold
Goldgraphene
F. Koppens et al, Nature Materials 2015R. Hillenbrand et al, Science, 2015 47
Comparison of plasmon figure of merits
T.Low, J.Caldwell, F. Koppens, L.M.Moreno, P. Avouris, T. Heinz et al, Nature Materials (2016)
2D materials has better FOM than 1nm Au in the infrared
48
Plasmonics beyond graphene
Phonon induced transparencySlow light
T.Low et al, Phys. Rev. Lett. (2014)H.Yan, T.Low et al, Nano Lett. (2014)
Anisotropic plasmonHyperbolic plasmon
T.Low et al, Phys. Rev. Lett. (2015)A.Nemilentsau, T.Low et al, Phys. Rev. Lett. (2016)
Non-reciprocal plasmonGiant Faraday rotation
A.Kumar et al, PRB Rapid. (2015)
49
From monolayer to bilayer graphene
1.0
1.5
2.0
2.5
µ
E
Con
duct
ivity
Re[
σ/σ 0]
k
E
phononinterband
T.Low, F.Guinea et al, Phys. Rev. Lett. (2014)
0.0 0.2 0.4 0.6 0.80.0
0.5
1.0
γ =0
Con
duct
ivity
Re[
Frequency ω (eV)
Pronounced plasmonic enhancements of IR phonon absorption
Narrow optical transparency window at zero detuning
Plasmon coupled to interband resonance 50
Observing phonon-induced transparency
10
15
Loss
Fun
ctio
n (a
.u.)
Higher dopingW = 100 nm
Chemical doping
T.Low, F.Guinea et al, Phys. Rev. Lett. (2014)
Substrate
Dielectric
SiO2/Si
Au
Vg
Experiments
1200 1500 1800 21000
5
Loss
Fun
ctio
n (a
.u.)
Frequency (cm-1)
1200 1500 1800 21000
5
10
15
1-T
/TS (
%)
Frequency (cm-1)
Higher doping
W = 100 nm Chemical doping
H.Yan, T.Low et al, Nano Lett. (2014)
plasmon phonon
From narrow absorption to narrow transparency 51
κ
Coupled two harmonic oscillators
1κ 2κ1 1,m γ 2 2,m γ
cos( )F tω
1 1 1 1 1 1 1 2
2 2 2 2 2 2 2 1
( ) cos( )
( ) 0
m x x x x x F t
m x x x x x
γ κ κ ωγ κ κ
+ + + − =+ + + − =
ɺɺ ɺ
ɺɺ ɺ
Equation of motion:
1 2γ γ≫
“plasmon” “phonon”
1.90 1.95 2.00 2.05 2.100.0
0.5
1.0
1.5
Pow
er P
1
Freq. ω
1 2ω ω>Narrow
absorption
1.51 2ω ω=Narrow
1 1 1
2 2 2
cos( )
cos( )
x a t
x a t
ω θω θ
= += +
Solutions:
Power absorption by “plasmon”:
1 1
11 1 12
( ) cos( ) ( )
( ) sin( )
P t F t x t
P t Fa
ωω θ
== −
ɺ
1.90 1.95 2.00 2.05 2.100.0
0.5
1.0
Pow
er P
1
Freq. ω
1 2ω ω=Narrow transparency
Destructive interference, mass becomes stationary
52
From induced-transparency to slow light
( ) ( ) ( )g
dnn n
dω ω ω ω
ω= +
( )( )g
g
cv
nω
ω=
( ) Re ( )n ω ε ω =
Highly dispersive n will yields Induced-transparency
Highly dispersive n will yields modified group velocity
A.H.Safavi-Naeini et al, Nature (2011)T.J.Kippenberg et al, Science (2008)
Slowing light for integrated photonic memory
53
Anisotropic 2D materials
Class of anisotropic materials
Highly anisotropic in-plane effective masses
Intraband optical process
Anisotropic intraband Drude conductivity
Interband optical process
Grp 5, e.g. black phosphorus1T TMD, e.g. ReS2, ReSe2
Transition metal trichalcogenides
L.Li et al, Nature Nano. (2014)S.Tongay et al, Nature Comms. (2014)
Interband conductivity also anisotropic
J.Qiao et al, Nature Comms. (2014)
Absorption edge at different energies for different polarization due to symmetry of the bands
54
Anisotropic 2D materials, optical conductivity
0
0g
g
σσ
σ
=
20
00
( ) lng
ie n is
i m
ω ωσ θ ω ωω η π ω ω
−= + − + + +
Graphene
Intraband Interband
0
0xx
yy
σσσ
=
2
( ) ln ,jjj j j
j j
ie n is j x y
i m
ω ωσ θ ω ω
ω η π ω ω −
= + − + = + +
Anisotropic semiconductor
Imaginary part follows from Kramers-Kronig
Im( ) Im( ) 0xx yyσ σ >
Im( ) Im( ) 0xx yyσ σ <Hyperbolic
Anisotropic
55
Anisotropic versus Hyperbolic plasmons
Anisotropic case Im( ) Im( ) 0xx yyσ σ > Hyperbolic case Im( ) Im( ) 0xx yyσ σ <
56
Hyperbolic plasmons, ray optics
Hyperbolic plasmon rays can be electrically control
A.Nemilentsau, T.Low, G.Hanson, Phys. Rev. Lett. (2016)57
Massive Dirac systems
Graphene/hBNR.V.Gorbachev et al, Science (2014)
Valley hall current Optical circular dichroism
TMDK.F.Mak et al, Nature Nano (2012)R.V.Gorbachev et al, Science (2014) K.F.Mak et al, Nature Nano (2012)H.Zeng et al, Nature Nano (2012)X.Xu et al, Nature Phys. (2014)
MoS2K.F.Mak et al, Science (2014)
( )1( )nE k e
v E kk
∂= − × Ω
∂
ℏ ℏ
These phenomena are static and dynamic manifestation of Berry physics
Berry curvature: can be viewed as an effective magnetic field due
to orbital part Bloch states
( ) ( )n nK KΩ = Ω −
( ) ( )n nK KΩ = −Ω −K and K’ valleys are related by time reversal symmetry
Spatial inversion symmetry requires thatBreaking spatial inversion
produces finite Berry curvature i.e. sublattice asymmetry
58
Massive Dirac system, optical conductivity
g xy
Kxy g
σ σσ
σ σ
= − '
g xy
Kxy g
σ σσ
σ σ−
=
22( ) ( )xy K K
ek k dkσ ρ= Ω∫ℏ
Optical pumping to create non-equilibrium valley imbalance
A.Kumar et al, PRB Rapid. (2015) 59
Non-reciprocal edge modes
2
0 20 0 0
22 0
( )g g xyi i k
c c
σ σ σε κκ ωε ε ε
+ ⋅ − + =
2 20k qκ ε= −
Bulk plasmons, continuous film
2 22
0
[3 2 2 sgn( )] | | 0g xyg xyq i q q
σ σσ σ
ωε ε +
− + + =
Edge plasmons, semi-infinite film
Linear dipole
Massive Dirac material
A.Kumar et al, PRB Rapid. (2015) 60
Isotropic plasmon(graphene)
bilayer
monolayer
Gold
Gold graphene
Experimental realization?
Graphene plasmons can be launched by nano Au optical
antenna, and mapped with SNOM
P.Alonso-Gonzalez et al, Science. (2014)
Chiral plasmon(gapped Dirac materials)
Hyperbolic plasmon(anisotropic materials)
61
Phonon-polaritons in boron nitrides
Optical mode
Acoustic mode
S.Dai et al, Science (2014)
62
hBN as natural hyperbolic material
2 2, ,
, , 2 2,
LO m TO mm m m
TO m mi
ω ωε ε ε
ω ω ω∞ ∞
−= +
− − Γ
hBN permittivity
Out-of-plane phonon modes
1 1, ,780cm , 830cmTO LOω ω− −= =
In-plane phonon modes
1 1, ,1370cm , 1610cmTO LOω ω− −= =
Elliptic Hyperbolic type I Hyperbolic type II 63
Phonon-polaritons in hBN
Dispersion within the Reststrahlen band
A.Kumar, T.Low, et al, Nano Lett. (2015)S. Dai et al, Science (2014)
A.Woessner, Nature Mat. (2014)
1 0( ) 2 tanhBN
q nt
εψω πε ψ
−
⊥
= − +
ψ ε ε⊥= ±
64
Plasmon-Phonon-polaritons in graphene-hBN
A.Kumar, T.Low, et al, Nano Lett. (2015)S. Dai et al, Science (2014)
A.Woessner, Nature Mat. (2014)
( )0 0 01 1 0( ) tan tanhBN
i q k Zq n
t
ε σ εψω πε ψ ε ψ
− −
⊥ ⊥
+ = − + +
( ) ( )( )( ) ( )( )
( )( )
0 0 0 0
0 0 0 0
0
0
1
1( ) ln
2 1
1hBN
i i q k Z
i i q k Ziq
t i
i
ψ ε ε σ εψ ε ε σ εψω
ψ ε εψ ε ε
⊥
⊥
⊥
⊥
− + + + = − × +
65
D. Basov et al, Science, 2014 J. Caldwell et al, Nature Comm. 2014 Hillenbrand et al, Nature Phot. 2015
Hyperbolic phonon polaritons in hBN
66
Hyperbolic polaritons beyond hBN
T.Low et al, Nature Materials (2016) 67
Exciton polaritons in 2D materials
T.Low et al, Nature Materials (2016) 68
Designers’ polaritons with 2D heterostructure
T.Low et al, Nature Materials (2016) 69
Acknowledgement
IBMHugen Yan, Marcus Freitag, Fengnian XiaWenjuan Zhu, Damon Farmer, Phaedon Avouris
SpainFrancisco Guinea, Luis Martin Moreno,Alexey Nikitin, Rafael Roldan, Frank Koppens
MIT – Nick FangU Wisconsin Milwaukee – George HansonNRL – Josh CaldwellNRL – Josh CaldwellStanford – Tony HeinzBrazil – Andrey Chaves
UMNRoberto Grassi, Eng Hock Lee, Yongjin Jiang, Kaveh Khaliji, SudiptaBiswas, Javad Azadani, Anshuman Kumar, Andrei Nemilentsau